right first
right
wrongUsing the First and Second Derivative Quiz
Question
Web resources available
This quiz tests the work covered in Lecture 17 and corresponds to Section 4.1
of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason,
McCallum et al.).
There are more web quizzes at Wiley, select Section 1.
You might want to go back to some of the resources for the second derivative if they were a bit hard at the time. They were http://archives.math.utk.edu/visual.calculus/3/graphing.14/ although some of the language is technical. There are also difficult, but useful quizzes at http://archives.math.utk.edu/visual.calculus/3/graphing.2/index.html and http://archives.math.utk.edu/visual.calculus/3/graphing.3/index.html.
There is some detailed information on this topic with exercises with fully worked solutions at http://www.math.ucdavis.edu/ kouba/CalcOneDIRECTORY/graphingdirectory/Graphing.html.
Question 1
Consider the function 
It has critical points at 
and
.
Differentiate 
and draw a sign diagram of 
to decide which one of the
following statements is true.
so our sign diagram looks like
This shows that
changes from positive to negative at 
and we can see
that means that there is a local maximum at 
does not change from negative to
positive.Choose numbers less than -2 and between -2 and 2 and substitute into
for
example 
and 
so 
changes from positive to negative.Question 2
Which of the following values for 
and 
give the function 
a local maximum at 
for the function
for the function 
is concave up everywhere, since 
so it cannot have
a local maximum.Let’s find the function where
has a critical value at
Since
we have 
and 
Now
and 
so 
Hence
Question 3
Consider the function 
Find the first and second derivatives of 
and
decide which one of the following statements is correct.
and so there is a critical point at 
and 
Therefore there is a local minimum at
is not a critical value.Question 4
Consider the function 
Find the first and second derivatives of 
and decide which of the following
statements are true.
For example, choice (a) should be false.
For example, choice (b) should be true.
 | = |  |
| = |  |
|
| = |  |
and 
Construct a sign diagram to determine the nature of these points
Hence there is a stationary points of inflection at
a local minimum at
and a local maximum at
For example, choice (c) should be true.
 | = |  |
 | = |  |
| = |  |
|
| = |  |
we have
Note that 
So there are points of inflection at
and 
For example, choice (d) should be false.
For example, choice (e) should be true.
For example, choice (f) should be false.
- False. You may have used the function to construct the sign diagram, instead of the derivative.
- True.This tells us that there are critical points at
= = = and
Construct a sign diagram to determine the nature of these points
Hence there is a stationary points of inflection at a local minimum at
and a local maximum at - True.Using the quadratic formula to find where
= = = = we have Note that
So there are points of inflection at and - False. You have made the same mistake as in b).
- True. This is true but there are more features.
- False. You can factorize the second derivative and get 3 points of inflection.